Yes, these are the two ways of describing the same baptism.

The first description is based on Matthew 28 where Jesus told his disciples

Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit. (Matthew 28:19)

Notice that it is one “name”, not three “names”. When Jesus went to heaven, he inherited his Father’s name:

Because of this, God raised him up to the heights of heaven and gave him a name that is above every other name. (Php 2:9)

In Acts, when the disciples baptised people, the shorter baptismal formula is used. For example

Acts 2:38 And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit.

Acts 10:48 And he commanded them to be baptized in the name of Jesus Christ.

Acts 19:5 On hearing this, they were baptized in the name of the Lord Jesus.

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